Gravitation

In general words; the force by which earth attracts all the objects towards it is called Gravitation. But, in scientific language; every object in the universe attracts every other object with an unseen force, and this force is called Gravitation or Force of Gravitation.

It was Great Issac Newton, English Scientist, who started thinking about the cause of falling of apple on him. He gave the theory about the Mystical Force of Gravitation, because of which everything fall on ground.

Example: When we throw anything upwards, it falls down on the earth.

Motion of Moon round the Earth

Moon revolves round the earth because of force of gravitation exerted by earth on moon. Force of Gravitation of earth pulls moon towards it.

The motion of moon around the earth is due to the centripetal force. The centripetal force is provided by the force of attraction of the earth. If there were no such force, the moon would pursue a uniform straight line motion.

Activity to show the motion of moon around the Earth

A piece, around one metre of strong thread or thin rope is taken.

A small piece of stone is tied at one end of the rope.

Rope is whirling round by holding other end in hand as shown in the figure.

By whirling, stone moves in a circular motion on circular path.

Now, rope is released from hand.

After releasing of rope, stone flies off along a straight line. This straight line is a tangent to the circular path.

Explanation:

Before the rope is released, the stone moves in a circular path with certain speed and changes direction involves change in velocity or acceleration. The force that causes this acceleration and keeps the body moving along the circular path is acting towards the centre. This force, which is acting towards the centre and keeps stone, moving in circular path, is called the centripetal force. Centripetal Force means centre seeking force.

In the absence of this centripetal force, the stone flies off along a straight line. This straight line is a tangent to the circular path.

In similar way, moon moves round the earth because of gravitational pull of earth towards its centre. This gravitational force acts similar to centripetal force which keeps moon bind on its orbit, i.e. on circular path.

Similar to motion of moon around the earth, all the planets revolves round the sun because Sun exerts gravitational pull to other planets.

The motion of all the planets round the sun takes place because all the objects in the universe attract each other.

This force by which all the objects in the universe attract each other is called the GRAVITATIONAL FORCE.

Tangent to a circle

A straight line that meets the circle at one and only one point is called a tangent to the circle. Straight line ABC is a tangent to the circle at point B.

Universal Law of Gravitation

Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centers of two objects.

Mathematical Formulation of Universal Law of Gravitation

Let A and B are two objects in the universe.

Let masses of A and B are M and m respectively.

Let the distance between centres of object A and B is equal to d.

Let the force of attraction between two objects be F.

Now, according to Universal Law of Gravitation,

(1) the force between two objects is directly proportional to the product of their masses.

That is, `F prop Mxxm` ---------- (i)

(2) And the force between two objects is inversely proportional to the square of the distance between them. That is

`F prop 1/d^2` ------------(ii)

Now, by combining equation (i) and (ii), we have

`F prop (Mxxm)/d^2` ----------- (iii)

`=>F=G(Mxxm)/d^2` -------------- (iv)

Where G is the constant of proportionality and is called the UNIVERSAL GRAVITATION CONSTANT.

By cross multiplication of equation (iv), we get

`Fxxd^2 = G\ Mxxm`

`=>G=(F\ d^2)/(Mxxm)` ------------- (v)

Equation (v) is known as the equation or formulae for UNIVERSAL GRAVITATION CONSTANT.

SI Unit of G (Universal Gravitation Constant)

SI unit of G can be obtained by substituting the SI units of Force, distance and mass.

The SI unit of Force = N (Newton)

The SI unit of distance = m (metre)

And the SI unit of mass = kg (kilogram)

Thus, SI unit of G `= N\ m^2\ kg^2`

The Value of G (Universal Gravitation Constant)

The value of G was found out by Henry Cavendish, a British scientist, also known for the discovery of Hydrogen, b using a sensitive balance.

The accepted value of G = 6.673 × 10^–11 N m² kg^–2.

Importance of the Universal Law of Gravitation

The universal Law of Gravitation successfully explained several phenomena which were believed to be unconnected.

(i) the force that bind us to the earth.

(ii) the motion of the moon around the earth

(iii) the motion of planets around the Sun.

(iv) the tides due to the moon and the Sun.

Force exerted by the earth on the Moon

Force exerted by the earth on the moon can be calculated using the formula of Universal Gravitational Constant, which is

`G = (F\ d^2)/(Mxxm)`

`=>F\ d^2= G(Mxxm)`

`=>F = G(Mxxm)/d^2` --------------- (iv)

[This formula has already been derived for Universal Gravitational constant]

Where, F = Force

G = Gravitational constant

M and m are masses of the given objects between which force of attraction is to be calculated.

And, d is the distance between given object.

Thus, by knowing mass of earth and moon and distance between them, the force exerted by earth on the moon can be calculated.

Experimentally we know that,

Mass of the earth (M) = 5.972 × 10²⁴ kg

⇒ M≈ 6 × 10²⁴ kg

Mass of the moon (m) = 7.4 × 10²² kg

Distance between earth and moon (d) = 3.84 × 10⁵ km

= 3.84 × 10⁵ × 1000 m

Or, d= 3.84 × 10⁸ m

Value of G = 6.7 × 10^–11 N m² kg^–2

Now by substituting the values of M, m, d and G in equation (iv)

`F = G(Mxxm)/d^2`

`=>F=2.01xx10^(20)N`

Thus, the force exerted by the earth on the moon = 2.01 × 10²⁰ N

Example Problem (1) Two objects of masses 500 kg and 50 kg lie at a distance of 300 m. Find the force of attraction between them. [Value of G = 6.7 × 10^–11 N m² kg^–2]

Solution:

Here, given, mass of one object (M) = 500 kg

Mass of second object (m) = 50 kg

Distance between the given two objects (d) = 300 m

Then, F =?

We know that, `F = G(Mxxm)/d^2`

Thus force exerted by one object on other = 1.86 × 10^–11 N Answer

Free Fall

Falling of an object towards earth under the force of gravitation alone; is called free fall.

Explanation of Free Fall

Earth attracts everything towards it. This happens because of force of gravitation. Whenever an object falls towards earth under the force of gravitation alone, it is said that the object is in free fall.

Is there any change of velocity of falling object while free fall?

While falling, there is no change in the direction of motion of the objects. And hence there is no change in the velocity of falling object while free fall.

But due the earth's attraction, there will be a change in the magnitude of the velocity. Any change in velocity involves acceleration, which is due to the earth's gravitational force.

Since the acceleration involves in the free fall is because of gravitational force of earth, thus is acceleration is called Acceleration Due to Gravitational Force of Earth or Acceleration due to gravity.

The acceleration due to gravity is denoted by the letter `g`.

Thus, SI unit of acceleration due to gravity (g) = m s^–2

Formulae of Acceleration Due to Gravity (g)

Newton's second law of motion says that Force is the product of mass and acceleration

i.e. F = m g ------------- (A)

Where, F = force, m = mass and g = acceleration.

And from the Universal Law of Gravitation, we know that,

`F=G(Mxxm)/d^2` ------------ (B)

Thus, from second Law of motion and Universal Law of Gravitation, i.e. from equation (A) and (B), we have

`mg = G(Mxxm)/d^2`

`=>g = GM/d^2` ------------- (C)

Where, M = Mass of the earth, d is the distance between the object and the earth and G is the Universal gravitational constant.

Equation (C), i.e. equation for the acceleration due to gravitation says that, acceleration due to gravitation (g) is free from the mass of the object while free fall.

In other words, if two object of different masses are set to free fall, then both of the object strike the earth at same time.

Acceleration due to gravitation for the object at the surface of earth

When object is at the surface of earth, then the distance between object and earth will be the distance between object and the centre of earth.

That means the distance `d` will be equal to `R` (Radius of Earth).

Thus by replacing `d` with `R` from equation (C), we get

`=>g = GM/R^2` ------------- (D)

Variation in `g` (Acceleration due to gravity) near the Pole and at the equator

Earth is not a perfect square. The radius of earth increases from pole to equator. Thus the value of `g` increases with decrease in value of `R` and the value of `g` decreases with increase in the value of `R`.

This means, the value of `g` becomes greater at the poles than at the equator, because, value of R is less at the Pole than the value of R at the equator of earth.

Calculation of The value of `g`

We know that, `=>g = GM/R^2` ------------- (D)

Thus, by substituting the value of M (Mass of earth), R(Radius of earth) and G(value of gravitational constant), the value of G can be calculated.

We know that,

G = 6.7 × 10^–11 N m² kg^–2

M = 6 × 10²⁴ kg

R = 6.4 × 10⁶ m

After substituting values of G, M and R in equation (D), we get

Thus, value of Acceleration due to Gravity (g) `= 9.8 m\ s^(-2)`.

For most of calculations, the value of `g` is taken constant on or near the earth.

Motion of Objects Under the Influence of Gravitational Force of the Earth

We know that, equation for `g=GM/R^2` or `g=GM/d^2`

Since, the value of `g` is free from mass of the object, thus, all object, whether big or small or any mass will fall at the same rate to the ground.

As `g` is constant near the earth, all the equation for the uniformly accelerated motion of objects become valid with acceleration (a) replaced with acceleration due to gravity (g).

This, means,

`v=u+at`

`s= ut + 1/2 at^2`

And `v^2 = u^2+2as` will becomes.

`v=u+g\ t`

`s= ut + 1/2 g\ t^2`

And `v^2 = u^2+2g\ s`.

In all the three equations, acceleration (a) has been replaced by acceleration due to gravity (g).

In all of the equations, `g` is taken as positive while object is falling towards earth and `g` is taken as negative while motion of object in upward direction, i.e. opposite to the gravity.